Proof of the Division Algorithm

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Discussion Overview

The discussion revolves around the application of the well ordering principle (WOP) in the context of the division algorithm, specifically whether it can be applied to subsets of non-negative integers. The scope includes conceptual clarification and informal debate regarding the definitions and implications of the WOP.

Discussion Character

  • Conceptual clarification, Debate/contested

Main Points Raised

  • One participant questions the applicability of the WOP to non-negative integers, suggesting a potential pedantic concern.
  • Another participant asserts that applying the WOP to a subset of non-negative integers is straightforward, as any subset of non-negative integers would also have a least member.
  • A later reply acknowledges a moment of confusion from the initial poster, indicating a realization of the correctness of the previous assertion.
  • Another participant humorously identifies with the notion of "senior moments," suggesting a shared experience among older individuals.

Areas of Agreement / Disagreement

The discussion appears to have a mix of agreement and light-hearted disagreement, with one participant initially questioning the application of the WOP and later conceding to the correctness of the other participant's explanation.

Contextual Notes

No specific limitations or unresolved mathematical steps are noted in the discussion.

Who May Find This Useful

Readers interested in number theory, particularly those exploring the foundations of mathematical principles like the well ordering principle and its applications.

matqkks
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In many books on number theory they define the well ordering principle (WOP) as:
Every non- empty subset of positive integers has a least element.
Then they use this in the proof of the division algorithm by constructing non-negative integers and applying WOP to this construction. Is it possible to apply the WOP to a subset of non-negative integers? Am I being too pedantic?
 
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It's rather obvious isn't it? "Applying the WOP to a subset of non-negative integers" would simply mean that, given X, a subset of the non-negative integers, any subset of X has a least member. And that is true because any subset of X is also a subset of the non-negative integers.

If that is not what you mean then please explain what you mean by "apply the WOP to a subset of non-negative integers".

 
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Yes of course. I just had a senior moment.
Thanks.
 
There are those of use who live in "senior moments"! We are called "seniors".
 
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